$\begingroup$What's wrong with the standard definitions available on wikipedia?$\endgroup$
– Tal FishmanOct 29 '11 at 23:02

$\begingroup$I am trying to look at the effects on the VIX index on hedge funds and I need to calculate the resulting skewness and Kurtosis when different weights of the VIX is added to the hedge fund portfolio. So the HF returns is considered as stock A and the Vix is considered as stock B. I have been using the matrices method to calculate the comoments. I need to find a formula to calculate the portfolio skewness and kurtosis.I have already calculated the skewness and kurtosis of each variable on their own. TY$\endgroup$
– user1642Nov 8 '11 at 15:48

$\begingroup$I am trying to look at the effects on the VIX index on hedge funds and I need to calculate the resulting skewness and Kurtosis when different weights of the VIX is added to the hedge fund portfolio. So the HF returns is considered as stock A and the Vix is considered as stock B. I have been using the matrices method to calculate the comoments. I need to find a formula to calculate the portfolio skewness and kurtosis.I have already calculated the skewness and kurtosis of each variable on their own. TY$\endgroup$
– user1642Nov 8 '11 at 15:49

2 Answers
2

"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is.

Skewness is defined as:

$E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$

and Kurtosis as:

$E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$

where X is your distro values (x_1, x_2, ... x_N), mean is the average of your distro values X (x_mean, a constant) and E[f(X)] is the Expectation of f(X) - i.e the mean of f(X).

So now you need to define your distributions. To be honest I don't know what the standards are for a given asset, but I imagine that if your asset price movements are ~ lognormal then you'll be wanting the daily (or whatever) percentage change in the value of the portfolio. These daily %age changes define your distribution X. Of course you'll need to consider how far back in time you go: 1 month data? 1 year?. So each daily %age change is your x_i. Calc the mean (probably close to zero), then your Skewness and Kurtosis per the formulas above.

$\begingroup$More specifically, skew (the third moment about the mean) is defined as $\int(x-\mu)^3 p(x)$ and kurtosis (the fourth moment about the mean) as $\int(x-\mu)^4 p(x)$.$\endgroup$
– strimp099Oct 29 '11 at 14:23

Assuming you have return time series
$$
r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T)
$$
for the 2 assets and asset weights $w_1$ and $w_2$, we can follow the calculation of the $N$-asset portfolio skewness laid out in another answer for a similar question.